{% extends 'homepage.html' %}
{% block content %}

{% if representation=='' %}
<p>
    The L-function $L(s,E)= \sum a_n n^{-s}$ of an Elliptic curve of
    {{ KNOWL('ec.q.conductor', title='conductor')}} $N$ has an 
    {{ KNOWL('lfunction.euler_product', title='Euler product')}}
    of the form
    \[
    L(s,E)=\prod_{p\mid N}  \left(1 - a_p p^{-s} \right)^{-1}\prod_{p\nmid N}  \left(1 - a_p p^{-s} + p^{-2s} \right)^{-1}
    \]
    and satisfies the {{ KNOWL('lfunction.functional_equation', title='functional equation')}}
    \[
    \Lambda(s,E)= N^{s/2}\Gamma_{\mathbb C}(s+1/2)\cdot L(s,E)= \Lambda(1-s,E).
    \]
</p>
{% else %}
<p>
    Given an L-function of an Elliptic curve of conductor $N$, the
    {{ KNOWL('lfunction.symm', title='symmetric n-th power $L$-functions')}}
    is defined by the {{ KNOWL('lfunction.euler_product', title='Euler product')}}
    \[
    L(s,E,\text{sym}^n)=\prod_{p\nmid N} \prod_{j=0}^n \left(1 - \frac{\alpha_p^{j} \beta^{n-j}_p}{p^s} \right)^{-1} \times \prod_{p|N} L_p(s)
    \]
</p>

{% endif %}

{{contents[0]|safe}}

{% endblock %}
